It is known that if either \(\beta \ge 2, 0 \le \gamma <1\)  or \(1 < \beta < 2, \gamma = 0\) , then there is a \(T_{\beta , \gamma }\) -invariant and ergodic measure \(\mu _{\beta , \gamma }\)  such that \(1/c \le \frac{d \mu _{\beta , \gamma }}{ d \lambda } \le c\)  for some \(c \ge 1\) . (See [1]} and [2]}.)
Also, it is known ([3]}) that \( h(T_{\beta , \gamma }) = \log \beta .\) 
On the other hand, Parry showed that if \(1 < \beta <2\)  and \(\gamma \ne 0\) , then \(T_{\beta , \gamma }\)  may not have an invariant measure, which is equivalent to the Lebesgue measure. (See [4]}). In what follows we will assume that either \(\beta \ge 2, 0 \le \gamma <1\) , or \(1 < \beta < 2, \gamma = 0\) .
